Matrices & Determinants Questions and Answers

Determine by inspection whether the given set of vectors is linearly dependent of independent Do NOT perform row reductions of a matrix Justify your answer 4 22 3

Determine whether B is the inverse of A A 4 3 5 4 O Yes O No B 4 3 54 Is B the multiplicative inverse of A

According to 2010 U S census the population of Fort Pierce was 41 797 The 2020 U S census found that the population of Fort Pierce had grown to 46 608 Use the exponential growth model Y Yekt where t is the number of years after 2010 to write an exponential growth function for the city of Fort Pierce Round the growth constant k to four significant figures The exponential growth function that models the population of Fort Pierce is Use the population growth model to predict the population of Fort Pierce in year 2021 Round the solution to the nearest whole number The model predicts that the population of Fort Pierce in 2021 will be Determine the year when model predicts the population of Fort Pierce will be 75 770 Round the solution to the nearest whole number According to the model the population of Fort Pierce will be 75 770 in the year

Fill in the blanks so that the resulting statements are true If A is an mxn matrix and B is an nxp matrix then AB is defined as an in matrix B matrix To find the product AB the number of nxp mxp nxm pxm nxn in matrix A must equal the number c

For y f x 6x 5 x 3 and Ax 4 find a Ay for the given x and Ax values b dy f x dx c dy for the given x and Ax values a Ay Round to four decimal places as needed

Let C and C be two biased coins such that the probabilities of getting head in a single toss are and 13 respectively Suppose a is the number of heads that appear when C is tossed twice independently and suppose is the number of heads that appear when C is tossed twice independently Then probability that the roots of the quadratic polynomial x ax are real and equal is C 1 1 2 A 40 81 B 20 81 WIN 1 D 4

Solve the system of equations using the inverse of the coefficient matrix of the equivalent matrix equation x y z 0 2y z 1 6 2x 3y The solution is Type an ordered thple Type an integer or a simplified fraction XX

2 Combine like terms in the following 2 4a 6 3a 10a

2 Let R Z6 x the polynomial ring with coefficients in Z6 Answer the following questions and give a short explanation for each answer a Is R a ring with unity b Is R an integral domain c Is R a finite ring d Does R contain any units If so list them all

1 Adam jogged 8 2 kilometers on Monday and 7 2 kilometers on Tuesday What is the difference between these two distances 2 On Monday George spent 10 hours studying On Tuesday he spent another 4 3 hours studying What is the combined time he spent studying 3 the A coach filled up a cooler with water until it weighed 14 pounds After the game cooler weighed 11 3 pounds How many pounds lighter was the cooler after the game 4 Carol s class recycled 5 boxes of paper in a month If they recycled another 8 4 boxes the next month was is the total amount they recycled 5 A king size chocolate bar was 117 inches long The regular size bar was 8 inches long What is the difference in length between the two bars 6 A small box of nails was 10 2 inches tall If the large box of nails was 6 2 inches taller how tall is the large box of nails 7 Lana had planned to walk 5 miles on Wednesday If she walked 3 2 miles in the morning how far would she need to walk in the afternoon 8 Mike bought a box of fruit that weighed 2 kilograms If he bought a second box that weighed 9 kilograms what is the combined weight of both boxes 9 While exercising Victor travelled 16 2 kilometers If he walked 10 2 kilometers and jogged the rest how many kilometers did he jog 10 Gwen bought a bamboo plant that was 3 8 feet high After a month it had grown another Who was the total height of the plant after a month 7 8 9 10

Cos 0 sin for the scalar variable 0 Find the characteristic polynomial ercise 12 3 4 Consider the matrix Re sin 0 Cos v choose 3 and solve for the the eigenvalues and try to solve for eigenvectors

Exercise 12 3 5 Let A V V be a linear map with eigen pair x A EV x R Prove the following A is an eigenvalue of A A 1 is an eigenvalue of A I where I V V is the identity map If A is invertible then X is an eigenvalue for A

10 5 2 Compute the characteristic polynomial and solve for th 17 18 4 genvalues For each eigenvalue find a basis for the eigenspace xercise 12 3 3 Consider the matrix A 8 2

xercise 12 3 2 Consider the matrix A 2 or each eigenvalue find a basis for the eigenspace 4 Compute the characteristic polynomial and solve for the eigenvalues

2a Twelve less than the quotient of x and 10

What is the horizontal asymptote of the following rational function x 8 2x 10 y O y 1 2 y 2 3 O y 0 There is no horizontal asymptote

which functions are invertible Select each correct answer S 10 8 6 4 10 8 6 4 2 s 8 do t 4 2 64 84 104 O of 6 the 2 60 61 Po N 12 6 8 10 9 x 8 10 X 10 8 6 10 8 6 4 2 2 10 8 6 4 2 2 4 6 8 10 64 8 do 8 6 4 No 6 10 XA K XA 6 8 10

x In4 5 equation below a Give a representation of the exact answer using logarithms Not equivalent 5 4 0 28 e 5x b Give a decimal approximation of the answer rounded to two decimal places Preview l n 4 5 x Preview x 0 28

Solve for y in the equation below If there is no solution enter DNE for each answer 6 3 9 39 a Give a representation of the exact answer using logarithms b Give a decimal approximation of the answer rounded to two decimal places

Solve for t in the equation below 8 80 a Give a representation of the exact answer using logarithms t b Give a decimal approximation of the answer rounded to two decimal places

6 Given A 2 3 1 27 30 6 1 1 1 0 1 4 0 compute sgn 4132 a1 0 1 a2 0 2 a3 0 3 a4 0 4

Find the solution of the system x y 1 mod 7 x z 2 mod 7 y z 3 mod 7

The economy of a small island nation is based on two sectors coal and steel Production of a dollar s worth of coal requires an input of 0 20 from coal and 0 25 from steel Production of a dollar s worth of steel requires an input of 0 56 from coal and 0 20 from steel Find the output from each sector that is needed to satisfy a final demand of 44 million for coal and 52 million for steel The output from the coal sector is approximately Round million

uppose A is a real matrix and A 832 84 42 Please find A

5 Find the singular value decomposition of A 1 1 Find the pseudo inverse A of matrix A 23 0 1 0 2 2 3

BY FComposition of the matrix ou must show the eigenvalues eigenvectors modal matrix and the diagonalization S AS Yo tailed procedure must be shown to receive any credits for the question arry out an eigen 1 1 2 A 0 3 0 Lo 1 2

For the function polynomial function x x x 1 x 2 Where does f crosses the y avis and touch th x axis Amo What is degree of AW clvl becomes larger 1 what function does

Suppose T is a transformation from R2 to R Find the matrix A that induces Tif T is a reflection over the line y 1 4x b rotation by 1 3 a A b A 00 00 00 00

a 1 0 0 0 1 0 0 0 a the following values can Ax attain for appropriate choices of a Select all that are correct Given the matrix A Cannot be determined from the information given 14 983 0 598 where a R which of

y 605 622 627 664 650 616 Jse exponential regression to find an exponential equation that best fits this data Give both a and b rounded o at least 4 decimal places Use linear regression to find an linear equation that best fits this data Give both m and b rounded to at least wo decimal places of these two which equation best fits the data Exponential

Refer to 11 12 What are the coordinates of each image a Ry axts 5 3 Rx 5 3 5 3 b Ryaz 1 6 Ry 196 13 Graph AXYZ with vertices X 2 4 Y 6 0 and Z 7 2 and Its Image after the composition Translation x y x 6 y Translation x y x 2y 3 Graph the image of the figure using the transformation 14 15 reflection across the x axis reflection across 16 Becky draws a triangle with vertices A 6 7 B 9 3 and C 4 2 on a coordinate grid She reflects the triangle across the line y 4 to get AA B C She then reflects the image across the line x 3 to get AA B C What are the coordinates of AA B C and AA B C 17 Graph the figure and its image under the given reflection Square JKLM with vertices J 4 6 K 0 6 L 0 2 and M 4 2 Ry arts 19 A point in Quadrant III is reflected over the y axis and then over the x axis What quadrant is it in now 18 ERROR ANALYSIS Jamil and Ashley are finding the coordinates of the image of 2 3 after a reflection in the x axis Is either of them correct EXPLAIN 1 P 2 3 2 x y x y 3 When reflecting 21 Graph ARST with vertices R 4 1 S 7 3 and T 64 and its image after the glide reflection TRANSLATION x y x y 1 REFLECTION In the y axis Jounill CQ 3 Ashley C 23 The Point 20 ERROR ANALYSIS Sasha says that for any AXYZ the reflection over the y axin composed with the reflection over the x axis is equivalent to a translation of AXYZ EXPLAIN Sasha s error

a J t sin Precipitation 12 Problem 2 The graph below models the monthly average precipitation in inches at a town on a particular mountain over a 3 year period where x is the number of months after April 2020 11 10 9 8 7 5 4 3 2 1 n 2 3 0 0 5 10 D J x 4 COS 2x 15 20 Month x 2 1 25 30 35

1 Jenna babysits for 15 per hour She also works as a tutor for 20 per hour She can work up to 13 hours this week and she wants to make at least 200 a Write a system of inequalities where x represents the number of hours she babysits and y represents the number of hours she tutors YA 20 19 label your axes and shade the 18 correct region 17 16 15 14 13 12 11 10 9 b Graph the system of inequalities Make sure to 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 c If she knows she has to babysit for 4 hours this week how many hours should she spend tutoring to make the most money Mark this point on your graph

Simplify the radical 9p 8 Select the correct choice below and fill in any answer boxes in your OA 9p 8 OB The root is not a real number A 4 Simplify your answer

1 3 Exponents Rules Simplify Your answer should contain only positive exponents 1 xy 2x y A x y C 3 2xy 2a a b 2ab A C 1663 a 1664 B D 1 B b 2a D I 2 4 2 a b 2ab 266 A C 2a b 2 m C A 4m n m m 11 mn 4 4n B D ab 2 8a b B n m D m 8

a Find the equation of the straight line which best fits the following points 1 1 2 3 4 6 and 5 6 b Let A 11 1 14 1 5 Find P the projection matrix on C A

A brewery produces regular beer and a lower carbohydrate light beer Steady customers of the brewery buy 11 units of regular beer and 15 units of light beer monthly While setting up the brewery to produce the beers the management decides to produce extra beer beyond that needed to satisfy customers The cost per unit of regular beer is 29 000 and the cost per unit of light beer is 35 000 Every unit of regular beer brings in 100 000 in revenue while every unit of light beer brings in 400 000 in revenue The brewery needs at least 12 000 000 in revenue and has determined that the total demand will be at least 45 units of beer Complete parts a and b a How much of each type of beer should be made so as to minimize total production costs units of regular beer and minimum total cost of Type integers or decimals units of light beer should be made for a

T R2 R is a linear transformation for which T A C E and and A is the standard matrix for this transformation then

Problem 7 Find the values of c c2 and c3 so that c 2 4 1 0 3 4 0 c3 1 0 0 3 12 3 Problem 7 enter the values of c c2 and c3 separated by commas

Question 7 5 5 9 Part 1 of 2 Use identities to find values of the sine and cosine functions of the function for the angle measure 2x given tan x 4 and cos x 0 cos 2x 0 Simplify your answer including any radicals Use integers or fractions for any numbers in the expression

5 COS Use an identity to simplify the expression Do not use a calculator T 2 9 sin 2 COS Sin 2 9 9 Type an exact answer using as needed Question 5 5 5

Let f x 5 The following table shows f x for various values of x f x 0 1 1 5 2 25 3 125 4 5 a Evaluate f 25 f 25 b Evaluate f 125 f 125 625 3125

Find the least squares solution of the system Ax b 2 1 2 1 1 3 12 0 X A 26 11 29 11 b

Calculate 1 3i Give your answer in a bi form Give the solution with smallest positive angle

3x An experimental drug changes a patient s temperature at the rate of x 5 degrees Fahrenheit per milligram of the drug where x is the amount of the drug administered Note The rate of change of temperature with respect to dosage is called the drug sensitivity Find the total change in temperature in F resulting from the first 5 milligrams of the drug Round your answer to three decimal places F

COS COS 7x 12 7T given below CTTP 2 Simplify your answer including any radicals Use integers or fractions for any numbers in the expression Rationalize all denominators

The width of a rectangle is x OA 3x 12x 27 ft OB 3x 12x 27 ft O C 3x 12x 27 ft O D 3x 10x 59 3 x 3 ft feet and its area is 3x 11x 23x 9 square feet Write a polynomial that represents the length of the rectangle m

43 Express the following using exponents 1282 204

f x x 3 and g x 5x find the following a fog x and its domain b gof x and its domain a fog x Simplify your answer

Let f x x 1 h x x 5 Find f o h 3 foh 3